The design of a fault detection and isolation (FDI) system for linear time-invariant processes subject to additive failures is considered. The paper first concentrates on the problem of residual generation. The redundancy relations are obtained by minimizing a quadratic cost function under non-convex quadratic inequality constraints. The constraints express, in some way, the desired performance of the FDI system, both for detection and isolation purposes. Robustness of the FDI system with respect to uncertainties in the model parameters can be achieved by representing the process behaviour with a finite set of models. Next, an approach to the design of a decision logic suitable for the proposed residual generation scheme is described. Finally, the feasibility of the method is illustrated with a simulation study of a FDI system for the actuator failures of a fluid mixer.